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Inmathematics, atrivial group orzero group is agroup that consists of a single element. All such groups areisomorphic, so one often speaks ofthe trivial group. The single element of the trivial group is theidentity element and so it is usually denoted as such:,, or depending on the context. If the group operation is denoted then it is defined by.
The similarly definedtrivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.
The trivial group is distinct from theempty set, which has no elements, hence lacks an identity element, and so cannot be a group.
Given any group, the group that consists of only the identity element is asubgroup of, and, being the trivial group, is called thetrivial subgroup of.
The term, when referred to " has no nontrivial proper subgroups" refers to the only subgroups of being the trivial group and the group itself.
The trivial group iscyclic of order; as such it may be denoted or. If the group operation is called addition, the trivial group is usually denoted by. If the group operation is called multiplication then can be a notation for the trivial group. Combining these leads to thetrivial ring in which the addition and multiplication operations are identical and.
The trivial group serves as thezero object in thecategory of groups, meaning it is both aninitial object and aterminal object.
The trivial group can be made a (bi-)ordered group by equipping it with the trivialnon-strict order.